Processes and devices for the thermal cutting of a running, meltable fabric run are known in from several sources, e.g. from WO-A-9103592 and EP-A-0389793. With these known processes and devices the meltable fabric run is cut by means of a hot cutting wire, the cutting wire being inclined to the fabric run at an angle of 90.degree. perpendicular to the cutting direction. This involves the disadvantage that due to the melted mass produced at the cutting point, relatively hard and rough melted edges are produced on thermal cutting. According to EP-A-0389793 the attempt was made to avoid these disadvantages in that the edge areas with the melted edges are folded over along folding lines parallel with the edges of the cut, whereafter the folding legs are pressed flat onto the adjoining surface zones of the material strip and fastened there by gluing. This procedure is relatively complicated, and an also relatively thick edge area ensues.
From EP-A-0152672 the technique of cutting a running, meltable fabric run, in the process of which the cut edge is furnished with a beveled cut surface, is known. This purpose is served by an ultrasonic cutting head with a V-shaped cutting element which has the form of an anvil and which interacts with an ultrasonic horn. The melting head is produced in the fabric material by friction between the anvil and ultrasonic horn. Irrespective of the fact that this cutting head is very complicated and susceptible to wear, and that it has a relatively large space requirement, the fabric material is displaced by the V-shaped anvil and produces large beads which adversely affect the appearance and the feel of the fabric material. In addition, the cutting operation nevertheless only conditionally produces edges resistant to fraying.